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Table 3 Maximum and mean errors for the linear mechanical and nonlinear models.

From: Data-guide for brain deformation in surgery: comparison of linear and nonlinear models

  Case 1 Case 2 Case 3 Case 4 Case 5 Case 6
Max error of the mechanical model Δ x = 3.0 m m Δ y = 3.3 m m Δ z = 1 .1 m m Δ x = 3.3 m m Δ y = 3.2 m m Δ z = 1.0 m m Δ x = 3.4 m m Δ y = 4.4 m m Δ z = 0 .3 m m Δ x = 2.9 m m Δ y = 3.1 m m Δ z = 0 .8 m m Δ x = 2.7 m m Δ y = 2.8 m m Δ z = 0 .4 m m Δ x = 4.0 m m Δ y = 3.4 m m Δ z = 1.7 m m
Mean error of the mechanical model Δ x = 1.2 m m Δ y = 1.2 m m Δ z = 0 .4 m m Δ x = 1.3 m m Δ y = 1.0 m m Δ z = 0 .3 m m Δ x = 1.3 m m Δ y = 1.7 m m Δ z = 0 .1 m m Δ x = 0.9 m m Δ y = 1. m m Δ z = 0 .6  m m Δ x = 0.9 m m Δ y = 0 .8 m m Δ z = 0 .1  m m Δ x = 1.0 m m Δ y = 0.7 m m Δ z = 0 .1 m m
Max error of the nonlinear model Δ x = 2.5 m m Δ y = 3.1 m m Δ z = 0 .3 m m Δ x = 3.0 m m Δ y = 3.1 m m Δ z = 0 .9 m m Δ x = 3.2 m m Δ y = 4.0 m m Δ z = 0 .3 m m Δ x = 2.9 m m Δ y = 2.8 m m Δ z = 0 .7 m m Δ x = 2.6 m m Δ y = 2.4 m m Δ z = 0 .4 m m Δ x = 3.0 m m Δ y = 2.9 m m Δ z = 1 .1 m m
Mean error of the nonlinear model Δ x = 0.8 m m Δ y = 1.1 m m Δ z = 0 .1 m m Δ x = 1.2 m m Δ y = 0.8 m m Δ z = 0 .2 m m Δ x = 1.1 m m Δ y = 1.3 m m Δ z = 0 .1 m m Δ x = 0.8 m m Δ y = 0.9 m m Δ z = 0 .4 m m Δ x = 0.7 m m Δ y = 0.7 m m Δ z = 0 .1 m m Δ x = 0.8 m m Δ y = 0.5 m m Δ z = 0 .9 m m
  1. Here, pre-operative and intra-operative MRI studies of six patients undergoing brain tumor surgery are used. The pre- and intra-operative images are registered rigidly, and then pairs of anatomical landmarks are determined by an expert radiologist in the corresponding images. One half of the points are used in the optimization process and the other half are used in the testing process. It should be mentioned that the overall mean of displacement in the x, y, and z direction for the linear mechanical model are 1.2, 1, and 0.3, respectively. These values for the nonlinear model are 0.9, 0.8, and 0.3, respectively. Based on the error of the testing landmarks, the proposed methods are evaluated and compared. Note that the nonlinear model in terms of both the maximum and the mean error in most cases has better results than the linear mechanical model.